I like the first best.
We just use a simple trick here. Reverse the sentence and you will see how easy it is toget the right answer. "I am better than he." Definitely not, "Me am better than he."
"He's better than me" is perfectly acceptable English usage/ Example: "He is better than me at remembering things."
"I am" is extraneous. "I" is hypercorrection.
As my English-teacher mom would have said, mockingly: "He's better than me am?"
RonBee you have to remember to always add the missing "am", so "He is better than me at remembering things" is grammatically incorrect because "He is better than me am" makes no sense. Thus, "He is better than I at remembering things" is correct
Says who? This is a made-up test that proves nothing. The majority of native speakers do not follow this pattern.